In this paper, certain system of linear homogeneous differential equations of second-order is considered. By using integral inequalities, some new criteria for bounded and L2 ½0;1Þ-solutions, upper bounds for values of improper integrals of the solutions and their derivatives are established to the considered system. The obtained results in this paper are considered as extension to the results obtained by Kroopnick (2014) [1]. An example is given to illustrate the obtained results.
Trang 1SHORT COMMUNICATION
On the boundedness and integration of
non-oscillatory solutions of certain linear
differential equations of second order
Department of Mathematics, Faculty of Sciences, Yu¨zu¨ncu¨ Yıl University, Kampus, 65080 Van, Turkey
A R T I C L E I N F O
Article history:
Received 14 February 2015
Received in revised form 6 April 2015
Accepted 13 April 2015
Available online 21 April 2015
Keywords:
Differential equation
Second order
Boundedness
L2½0; 1Þ-solutions
Non-oscillatory
A B S T R A C T
In this paper, certain system of linear homogeneous differential equations of second-order is considered By using integral inequalities, some new criteria for bounded and L 2
½0; 1Þ-solu-tions, upper bounds for values of improper integrals of the solutions and their derivatives are established to the considered system The obtained results in this paper are considered as exten-sion to the results obtained by Kroopnick (2014) [1] An example is given to illustrate the obtained results.
ª 2015 Production and hosting by Elsevier B.V on behalf of Cairo University.
Introduction
Very recently, Kroopnick[1]discussed some qualitative
prop-erties of the following scalar linear homogeneous differential
equation of second order
He established sufficient conditions under which all solutions of
Eq.(1) are bounded, and the solution and its derivative are both elements in L2½0; 1Þ Furthermore, the author proved that when the solutions are non-oscillatory, they approach 0
as t! 1 and calculated upper bounds for values of improper integrals of the solutions and their derivatives, that is, for
R1
0 x2ðsÞds andR1
0 ½x0ðsÞ2ds:Finally, Kroopnick[1]introduced
a short discussion about the L2½0; 1Þ-solutions to second order scalar linear homogeneous differential equation x00þ qðtÞx ¼ 0 The results obtained by Kroopnick are summarized in Theorems A and B
Theorem A (Kroopnick [1, Theorem I]) Given Eq (1) Suppose að:Þ is a positive element in C½0; 1Þ such that
A0> aðtÞ > a0>0 for some positive constants A0 and a0,
* Corresponding author.
E-mail address: cemtunc@yahoo.com (C Tunc¸).
Peer review under responsibility of Cairo University.
Production and hosting by Elsevier
Journal of Advanced Research (2016) 7, 165–168
Cairo University Journal of Advanced Research
http://dx.doi.org/10.1016/j.jare.2015.04.005
2090-1232 ª 2015 Production and hosting by Elsevier B.V on behalf of Cairo University.
Trang 2then all solutions to Eq (1)are bounded Moreover, if any
solution xð:Þ is non-oscillatory, then both xðtÞ ! 0 and
x0ðtÞ ! 0 as t ! 1 Finally, the solution and its derivative
are both elements of L2½0; 1Þ
The second result proved by Kroopnick[1]is the following
theorem
Theorem B (Kroopnick [1, Theorem II]) Under the
condi-tions of Theorem I, the following inequalities hold:
Z 1
0
½x0ðsÞ2ds 6½x0ð0Þ2þ k2½xð0Þ2
2a0 and
Z t
0
½x0ðsÞ2ds 6 xð0Þx0ð0Þ þ 1
2k2að0Þ½xð0Þ2
þ½x
0ð0Þ2þ k2½xð0Þ2 2a0k2 :
It should be noted that Kroopnick [1] proved both of
Theorems A and B by the integral inequalities
In this paper, in lieu of Eq.(1), we consider the more
gen-eral vector linear homogenous differential equation of the
sec-ond order of the form
where X2 Rn
; t2 Rþ; Rþ¼ ½0; 1Þ; að:Þ; bð:Þ : Rþ! ð0; 1Þ
are continuous functions and að:Þ and bð:Þ have also lower
and upper positive bounds
It should be noted that Eq.(2)represents the vector version
for the system of real second order linear non-homogeneous
differential equations of the form
x00þ aðtÞx0
iþ bðtÞxi¼ 0; ði ¼ 1; 2; ; nÞ:
Then, it is apparent that Eq.(1)is a special case of Eq.(2)
It is worth mentioning that, in the last century, stability,
instability, boundedness, oscillation, etc., theory of differential
equations has developed quickly and played an important role
in qualitative theory and applications of differential equations
The qualitative behaviors of solutions of differential equations
of second order, stability, instability, boundedness, oscillation,
etc., play an important role in many real world phenomena
related to the sciences and engineering technique fields See,
in particular, the books of Ahmad and Rama Mohana Rao
[2], Bellman and Cooke [3], Chicone [4], Hsu [5],
Kolmanovskii and Myshkis [6], Sanchez [7], Smith [8],
Tennenbaum and Pollard[9] and Wu et al.[10] In the case
n¼ 1; aðtÞ ¼ 0 and bðtÞ – 0, Eq.(2)is known as Hill equation
in the literature Hill equation is significant in investigation of
stability and instability of geodesic on Riemannian manifolds
where Jacobi fields can be expressed in form of Hill equation
system[11] The mentioned properties have been used by some
physicists to study dynamics in Hamiltonian systems[12] Eq
(1)is also encountered as a mathematical model in
electrome-chanical system of physics and engineering [2] By this, we
would like to mean that it is worth to work on the qualitative
properties of solutions of Eq.(2)
In this paper, stemmed from the ideas in Kroopnick[1,13],
Tunc [14,15] and Tunc and Tunc [16], etc., we obtain here
some new criteria related to the bounded and L2½0;
1Þ-solutions, upper bounds for values of improper integrals of solutions of Eq.(2)and their derivatives, where the functions að:Þ and bð:Þ do not need to be differentiable at any point and the Gronwall inequality is avoided which are the usual cases The technique of proofs involves the integral test and
an example is included to illustrate the obtained results This work has a new contribution to the topic in the literature This case shows the novelty of this work The results to be established here may be useful for researchers working on the qualitative theory of solutions of differential equations
The main results
In this section, we introduce the main results We arrive at the following theorem:
Theorem 1 Given Eq (2) Suppose að:Þ and bð:Þ are positive elements in C½0; 1Þ such that
A0> aðtÞ > a0>0 and B0> bðtÞ > b0>0 for some positive constants A0; a0; B0 and b0and for all t2 Rþ Then all solutions ofEq.(2)are bounded Moreover, if any solution Xð:Þ of Eq.(2)is non-oscillatory, then both XðtÞk k ! 0 and kX0ðtÞk ! 0 as t ! 1 Finally, the solution and its derivative are both elements of L2½0; 1Þ
Proof First, we prove boundedness of solutions of Eq (2) When we multiply Eq.(2)by 2X0ðtÞ, it follows that
2 Xh 0ðtÞ; X00ðtÞi þ 2 aðtÞXh 0ðtÞ; X0ðtÞi þ 2 bðtÞXðtÞ;Xh 0ðtÞi ¼ 0: ð3Þ Integrating estimate(3)from 0 to t and then applying integra-tion by parts to the first term on the left hand side of(3), we find
2
Z t 0
X0ðsÞ; X00ðsÞ
Z t 0
aðsÞX0ðsÞ; X0ðsÞ
þ 2
Z t 0
bðsÞXðsÞ; X0ðsÞ
and
X0ðtÞ
k k2 Xk 0ð0Þk2þ 2
Z t 0
aðsÞ Xk 0ðsÞk2ds
þ 2
Z t 0
bðsÞXðsÞ; X0ðsÞ
respectively
In view of the last two terms included in estimate(4), first apply the mean value theorem for integrals and then use the assumptions of Theorem 1, it follows that
2
Z t 0
aðsÞ Xk 0ðsÞk2ds¼ 2aðtÞ
Z t 0
X0ðsÞ
k k2dsP 2a0
Z t 0
X0ðsÞ
k k2ds;
2
Z t 0
bðsÞXðsÞ; X0ðsÞ
Z t 0
XðsÞ; X0ðsÞ
¼ 2bðtÞ
Z t 0
XðsÞ; X0ðsÞ
P bkXðtÞk2 bkXð0Þk2;
Trang 3where 0 < t< t.
On gathering the obtained estimates in(4), we have
2a0
Z t
0
X0ðsÞ
k k2dsþ Xk 0ðtÞk2 Xk 0ð0Þk2þ b0kXðtÞk2
b0kXð0Þk26kX0ðtÞk2 Xk 0ð0Þk2þ 2
Z t 0
aðsÞ Xk 0ðsÞk2ds
þ 2
Z t
0
bðsÞXðsÞ; X0ðsÞ
Hence, in view of(4)and the last estimate, it is obvious that
2a0
Z t
0
X0ðsÞ
k k2dsþ Xk 0ðtÞk2þ b0kXðtÞk2
It follows from estimate(5)that all terms on the left hand side of
(5)are positive and the right hand side of (5)is bounded as
t! 1 Hence, we can conclude that both XðtÞk k and Xk 0ðtÞk
must remain bounded when t! 1 Otherwise, the left hand side
of(5)would become infinite, which is impossible Furthermore,
it can be seen from(5)that Xk 0ð:Þk is in L2½0; 1Þ since the integral
Rt
0kX0ðsÞk2dsmust be bounded when t! 1
Next, we show that Xð:Þk k, too, is in L2½0; 1Þ if the solution
is non-oscillatory Multiply Eq.(2)by XðtÞ and integrate from
0 to t and then integrate the first term by parts to obtain the
following estimate
XðtÞ; X0ðtÞ
h i Xð0Þ; Xh 0ð0Þi
Z t 0
X0ðsÞ; X0ðsÞ
þ 2
Z t
0
aðsÞX0ðsÞ; XðsÞ
Z t 0
bðsÞXðsÞ;XðsÞ
Applying the mean value theorem for integrals to the fourth
and fifth terms on the left hand side of(6), we have
XðtÞ; X0ðtÞ
h i Xð0Þ; Xh 0ð0Þi
Z t 0
X0ðsÞ; X0ðsÞ
h ids þ 2aðtÞ
Z t
0
X0ðsÞ; XðsÞ
h ids þ 2bðtÞ
Z t 0
XðsÞ; XðsÞ
so that
XðtÞ; X0ðtÞ
Z t
0
X0ðsÞ
k k2dsþ aðtÞ XðtÞk k2þ 2bðtÞ
Z t
0
XðsÞ
k k2ds¼ aðtÞ Xð0Þk k2þ Xð0Þ; Xh 0ð0Þi; ð7Þ
where 0 < t< t
Now, if we can show that X0ð:Þ eventually does not change
sign, then Xð:Þ must eventually be monotonic We will then
show that kXð:Þk is also an element of L2½0; 1Þ These two
facts imply along with what has been proven before will show
that both kXðtÞk and kX0ðtÞk must approach 0 as t ! 1
Otherwise, the L2½0; 1Þ-convergence of the solution and its
derivative could not occur We assume that X0ð:Þ does change
sign infinitely often, then it is oscillatory Consequently,
X0ðtÞ ¼ 0 infinitely often However, this means that if
XðtÞ > 0, then X00ðtÞ < 0 So, XðtÞ has an infinite number of
consecutive critical points which are all relative maxima which
is impossible Likewise, if XðtÞ < 0, then we have an infinite
number of consecutive relative minima which is also
impos-sible Consequently, X0ðtÞ must be non-oscillatory This
implies that XðtÞ; Xh 0ðtÞi does not change sign In view of estimate(7), it follows that since the right hand side of (7)is both positive and bounded and all terms on the left hand side
of(7)are either positive or bounded, we may conclude that Xð:Þ
k k is in L2½0; 1Þ and therefore both kXðtÞk ! 0 and
X0ðtÞ
k k ! 0 as t ! 1
The proof is complete h The second main result of this paper is the following theorem:
Theorem 2 If the conditions of Theorem 1 hold, then the following estimates are satisfied:
Z 1 0
X0ðsÞ
k k2ds 6 1
2a0 X
0ð0Þ
k k2þ b0
and
Z 1 0
X0ðsÞ
k k2ds 61
2
aðtÞ bðtÞkXð0Þk
2
bðtÞ Xð0Þ; X
0ð0Þ
þ1 2
aðtÞ bðtÞkXðtÞk
2
2a0bðtÞ X
0ð0Þ
þ b0 2a0bðtÞkXð0Þk
2
Proof Consider estimate(5), that is, 2a0
Z t 0
X0ðsÞ
k k2dsþ Xk 0ðtÞk2þ b0kXðtÞk2
6kX0ð0Þk2þ b0kXð0Þk2: Taking into consideration the assumptions of Theorem 2, we can conclude that
XðtÞ
k k ! 0 and kX0ðtÞk ! 0 as t ! 1:
Then after dividing both sides of last estimate by 2a0,it can be followed that
Z 1 0
X0ðsÞ
k k2ds 6 1
2a0 X
0ð0Þ
k k2þ b0
2a0kXð0Þk2 as t! 1: Thus, estimate(8)easily follows
To arrive at estimate(9), we rearrange estimate(7)as
XðtÞ; X0ðtÞ
2 ðtÞ XðtÞk k2þ bðtÞ
Z t 0
XðsÞ
k k2ds
61
2aðtÞ Xð0Þk k2þ
Z t 0
X0ðsÞ
k k2dsþ Xð0Þ; Xh 0ð0Þi: ð10Þ
Let t! 1 By removing the positive terms XðtÞ; Xh 0ðtÞi and
1aðtÞ XðtÞk k2 in estimate(10) and using estimate(8), we can write from(10)that
bðtÞ
Z 1 0
XðsÞ
k k2ds 61
2 ðtÞ Xð0Þk k2þ
Z 1 0
X0ðsÞ
k k2ds
þ Xð0Þ; Xh 0ð0Þi
61
2 ðtÞ Xð0Þk k2þ Xð0Þ; Xh 0ð0Þi
þ1
2 ðtÞ XðtÞk k2þ 1
2a0 X
0ð0Þ
þ b0 2a kXð0Þk2:
Trang 4Finally, when we divide last estimate by bðtÞ, we obtain
esti-mate(9) The proof of Theorem 2 is now complete h
Example Let n¼ 2 Consider non-homogeneous linear
dif-ferential system given by
x00
1
x00
2
t2þ 1
1
x0 2
þ ð2 þ expðtÞÞ x1
x2
0
; tP 0:
ð11Þ Here,
aðtÞ ¼ 1 þ 1
t2þ 1; bðtÞ ¼ 2 þ expðtÞ:
Then, we have
a0¼1
2< aðtÞ ¼ 1 þ 1
t2þ 1<3¼ A0;
b0¼ 1 < bðtÞ ¼ 2 þ expðtÞ < 4 ¼ B0:
qð:Þ 2 L1ð0; 1Þ Hence, all the conditions of Theorems 1 and 2
hold to system(11)
Remark Kroopnick[1]proved Theorem A and Theorem B by
the integral test to scalar linear homogenous differential
equation of second order, x00þ aðtÞx0þ k2x¼ 0; ðk 2 RÞ In
defiance of the results of Kroopnick[1], which are not new,
the proofs presented in[1]are new and simplify some previous
related works in the literature since the Gronwall inequality is
avoided and að:Þ does not need to be differentiable at any point,
which are the usual cases It should be noted that the equation
discussed in [1] is a special case of our equation
X00þ aðtÞX0þ bðtÞX ¼ 0 When we take n ¼ 1, then Eq (2)
and the assumptions of Theorems 1 and 2 reduce to those of
Kroopnick [1, Theorem 1, Theorem 2] Since the Gronwall
inequality is avoided and að:Þ and bð:Þ do not need to be
dif-ferentiable, the proofs of this paper are new and the results of
this paper simplify previous works in the literature (see[1])
Furthermore, our results extend the results of Kroopnick[1,
Theorem 1, Theorem 2]and that in the literature
Conclusion
A linear homogeneous differential system is considered Some
sufficient conditions are established which guarantee to the
bounded and L2½0; 1Þ-solutions, give upper bounds for values
of improper integrals of the solutions and their derivatives for
the considered system To prove the main results, we benefited
from well-known integral inequalities The results obtained
essentially complement and extend some known results in the
literature An example is introduced to illustrate the main
results of this paper
Conflict of interest
The authors have declared no conflict of interest
Compliance with Ethics Requirements This article does not contain any studies with human or animal subjects
Acknowledgment The authors of this paper would like to expresses their sincere appreciation to the anonymous referees for their valuable com-ments and suggestions which have led to an improvement in the presentation of the paper
References
[1] Kroopnick Allan J On the integration of L 2 -solutions of non-oscillatory solutions to x 00 þ aðtÞx 0 þ k 2 x ¼ 0 Int Math Forum 2014;9(10):475–81
[2] Ahmad S, Rama Mohana Rao M Theory of ordinary differential equations With applications in biology and engineering New Delhi: Affiliated East-West Press Pvt Ltd.; 1999
[3] Bellman Richard, Cooke Kenneth L Modern elementary differential equations Reprint of the 1971 2nd ed New York: Dover Publications, Inc.; 1995
[4] Chicone C Ordinary differential equations with applications Texts in applied mathematics, 2nd ed., vol 34 New York: Springer; 2006
[5] Hsu Sze-Bi Ordinary differential equations with applications Series on applied mathematics, 2nd ed., vol 21 Hackensack (NJ): World Scientific Publishing Co Pte Ltd.; 2013 [6] Kolmanovskii V, Myshkis A Introduction to the theory and applications of functional differential equations Dordrecht: Klu wer Academic Publishers; 1999
[7] Sanchez David A Ordinary differential equations and stability theory An introduction Republ of 1968 orig publ by Freeman and Company Dover Books on Advanced Mathematics, vol.
VI New York: Dover Publications, Inc.; 1979 [8] Smith Hal An introduction to delay differential equations with applications to the life sciences Texts in applied mathematics, vol 57 New York: Springer; 2011
[9] Tennenbaum M, Pollard H Ordinary differential equations: an elementary textbook for students of mathematics, engineering, and the sciences New York: Dover Publications; 1985 [10] Wu Min, He Yong, She Jin-Hua Stability analysis and robust control of time-delay systems Berlin: Springer-Verlag; 2010 [Science Press Beijing, Beijing]
[11] Gallot S, Hulin D, Lafontaine J Riemannian geometry Universitext Berlin: Springer-Verlag; 2004
[12] Pettini M, Valdettaro R On the Riemannian description of chaotic instability in Hamiltonian dynamics Chaos 1995;5(4):646–52
[13] Kroopnick Allan J Two new proofs for the boundedness of solutions to x 00 þ aðtÞx ¼ 0 Missouri J Math Sci 2013;25(1):103–5 [14] Tunc C Boundedness results for solutions of certain nonlinear differential equations of second order J Indones Math Soc 2010;16(2):115–26
[15] Tunc C A note on the bounded solutions to
x 00 þ cðt; x; x 0 Þ þ qðtÞbðxÞ ¼ fðtÞ Appl Math Inform Sci (AMIS) 2014;8(1):393–9
[16] Tunc C, Tunc E On the asymptotic behavior of solutions of certain second-order differential equations J Franklin Inst 2007;344(5):391–8